General natural riemannian almost product and parahermitian structures on tangent bundles drutaromaniuc, simonaluiza, taiwanese journal of mathematics, 2012. Complex vector bundle vs holomorphic vector bundle vs. A nonzero vector bundle is called decomposable if it is the whitney sum of two nonzero vector bundles. A complex vector bundle is a holomorphic vector bundle if x is a complex manifold and if the local. Deformations of tensor structures on a normal tubular neighborhood of a submanifold 1. Denote by a p, q m, e the space of smooth sections of the bundle of p, q forms valued in e. Show that the nontrivial vector bundle on s1 constructed above is not orientable. A complex vector bundle is a fiber bundle with fiber cn and structure group gln, c. In this context h can be regarded as a jinvariant complex symmetric bilinear form on the complexi cation of the almost complex vector bundle tm, and. Let e be a nef vector bundle on a compact complex connected. As the differential geometric counterpart to the stability, i introduced the concept of an einsteinhermitian vector bundle. Almost complex structures and geometric quantization 3 where tdx is the todd class of x. Ramanan no part of this book may be reproduced in any form by print, micro. Show that if m has an almost complex structure then it must be evendimensional.
We show that in finitedimensional nonlinear approximations, the best rterm approximant of a function f almost always exists over c but that the same is not true over r, i. By definition, a complex structure is a bundle map between a real vector bundle e and itself. Given a lie group g, a principal g bundle over a space bcan be viewed as a parameterized family of spaces f x, each with a free, transitive action of gso in particular each f x is homeomorphic to g. Roughly speaking, a complex structure on v enable us to \multiply p 1 on v and thus convert v into a complex vector space. An almost complex structure j on m is a linear complex structure that is, a linear map which squares to. The real linear map jextends to a complex linear map jon this space, and since j2 id. An almost complex structure on a 4dimensional oriented vector space is called orientation compatible if there is an oriented basis of the form e. Vector bundles in algebraic geometry enrique arrondo notes prepared for the first summer school on complex geometry villarrica, chile 79 december 2010 1. We will generalize the standard one to the new ones such that the induced 0, 2tensor on the tangent bundle using these structures and liouville 1form will be a riemannian metric.
This paper originates from the talk almost complex structures on. Complex geometry is a highly attractive branch of modern mathematics that has witnessed many years of active and successful research and that has recently obtained new impetus from physicists interest in questions related to mirror symmetry. Any complex vector bundle over a paracompact space admits a hermitian metric. Old and new structures on the tangent bundle munteanu, marian ioan, 2007.
Such a j is called an almost complex structure and makes the real tangent bundle into a complex vector bundle via declaring iv jv. Such a manifold mis called an almost complex manifold. Curvature of a complex line bundle and hermitian line bundle. For p2m we consider the complexi ed tangent space t pm c, which is a complex vector space of dimension dimmand a real vector space of dimension 2dimm. Complex algebraic geometry 5 is the clinear extension of f. If x is a complex manifold, then the total space of the complex vector bundle t1, 0x. A real oriented vector bundle e is said to admit an almost complex structure if there is a complex vector bundle f such that. A complex vector bundle is a fiber bundle with fiber cn and structure group gln,c. It is well known that the tensor bundle of type 1, q of m admits an almost complex structure on the pure crosssection, if m admits an almost complex structure.
The dolbeault complex may be replaced with the rolled up version 1. Then metrics always exist, and every sympletic vector bundle is a complex vector bundle and vice versa. More precisely, let ebe a complex vector bundle over an almost complex manifold x,j. Due to its interactions with various other fields differential. Vector bundles and connections universiteit utrecht. Almost complex structures on connected sums of complex projective. On the geometry of tangent bundles induced by the isotropic. Let f be a real vector space and let w be a complex. For example, gxw snx is weakly almost complex for all n but is almost complex only when n 3 or 7 1. Michele audin, symplectic and almost complex manifolds.
V v is an almost complex structure, then dimrv is even and v admits, in a natural way, the structure of a complex vector space. Equivalently, each tangent space t pmis equipped with a complex structure j pwhich varies smoothly with respect to p. Analysis on almost complex manifolds almost complex manifolds. Characterizing moisezon spaces by almost positive coherent. But one suspects that this may not be possible, in the same general way that. Complex vector bundle vs holomorphic vector bundle vs almost. A generalized almost complex structure is a choice of a halfdimensional isotropic subspace of each fiber of the direct sum of the complexified tangent and cotangent bundles.
A connection of type 1, 0 on e is a differential operator. Analysis on almost complex manifolds almost complex manifolds suppose m. There is a bijection between bundle almost complex structures and. For x a manifold we say that x has an almost complex structure respectively stable almost complex structure if. J, e of order 1 acting on the complex a, and satisfying the following two. We will focus on complex vector bundles, but for almost everything a real. Recall that a complex structure on a real vector space v is automorphism j. This paper discussed vector bundle and complex line bundle,which is a special case of a. Almost complex structures and geometric quantization. A complex vector bundle can be thought of as a real vector bundle with an additional structure, the complex structure. More generally, if xis any space, we can identify the homology groups e nxwith bordism groups of stably almost complex nmanifolds equipped with a map to x. Each j pmakes t pminto a complex vector space, and with j, tmbecomes a complex vector bundle. Introduction an almost complex structure on an even dimensional smooth manifold m is a smooth linear vector bundle map j on the tangent bundle of m satisfying j2.
A generalized almost contact structure on mis a sub bundle eof e1m c such that eis isotropic and e1m. Symplectic linear group and linear complex structures15 4. The main purpose of this paper is to investigate a similar problem for tensor bundles p tmq of type, p q, p 1. A bundle almost complex structure j on e is a special almost complex structure on the total space of the e, see 5,6. Vanishing theorems for the halfkernel of a dirac operator 5 suppose e is a cli. Embeddings of almost hermitian manifold in almost hyper. They show that there is a bijection between bundle almost complex structures and the pseudoholomorphic structures on esee proposition 4. This assertion, while elementary, is fundamental in the arguments that follow, so we will prove it. Atiyahs classification of vector bundles on elliptic curves.
The existence of an almost complex structure j on a manifold m is. An almostcomplex manifold is a smooth real manifold j tm jx ix m. Whenever a is a complex manifold, the complexified tangent bundle splits into type components. The latter, viewed as a complex vector bundle using j.
Pdf almost complex structures on spheres researchgate. Now suppose that xhas merely an almost complex structure and lis a hermitian line bundle with compatible hermitian connection. An almostcomplex manifold is a smooth real manifold j tm. Let e be a smooth complex vector bundle over an almost complex manifold m, j of complex dimension n. If x is a complex manifold, then the total space of the complex vector bundle t1,0x. The space of compatible almostcomplex structures on m. A read is counted each time someone views a publication summary such as the title, abstract, and list of authors, clicks on a figure, or views or downloads the fulltext. An almost complex structure on m is a fiberwise complex struc ture on the tangent bundle tm, i. A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank saji, kentaro and tsuchida, asahi, rocky mountain journal of mathematics, 2019. Complex vector bundles and complex manifolds definition 1. A vector bundle e over a complex space x is called semipositive if the following two conditions are satisfied. An almost complex structure on v is a linear transformation j. They may also be expressed as moduli stacks of almost complex structures, see here. The basic invariant of a complex vector bundle is a chern class.
We introduce connection and curvature of complex line bundle and prove that any curvature 2form for a complex line bundle is both closed an integral. This concept has been generalized to vector bundles and, more generally, coherent sheaves over algebraic manifolds by takemoto, bogomolov and gieseker. The only two vector bundles with base space a circle and onedimensional. Since tx is not complex manifold, how to construct a holomorphic vector bundle. Lecture notes geometry of manifolds mathematics mit. Vector bundles thus combine topology with linear algebra, and the study of vector bundles could be called linear algebraic topology.
Complex manifolds stefan vandoren1 1 institute for theoretical physics and spinoza institute utrecht university, 3508 td utrecht, the netherlands s. A generalized almost contact structure on mis a subbundle eof e1m c such that eis isotropic and e1m. The linear algebra introduced above may be applied pointwise to the tangent bundle of m. Differential geometry of generalized lagrangian functions okubo, katsumi, journal of mathematics of kyoto university, 1991. The \holomorphic tangent bundle of a complex manifold recall that for any complex manifold x, there is a decomposition of the complexi ed tangent bundle. Another question, when does a real manifold carry a complex structure so that it is indeed a complex manifold. If the dimension of the vector space is mthen the bundle is often called an mplane bundle. Let ebe a complex vector bundle with a pseudoholomrphic structure over a compact almost complex manifold x,j, then h0x,e is. Holomorphic vector bundles northwestern university. By construction, trivial vector bundles are always orientable we can construct them from glkcocycles whose maps are constant, valued at the identity map. A new class of almost complex structures on tangent bundle. First, consider the integrability condition for almost complex structures on a. The curvature fl of rl is an imaginary valued 2form on m.
Xis a complex topological vector bundle where the total space ehas been endowed with a complex structure making the projection. The linear algebra introduced above may be applied pointwise to the tangent. An almost complex manifold is a ck real 2nmanifold together with a ck endomorphism iof its tangent bundle i. Almost complex manifolds an almostcomplex manifold is a smooth real manifold m equipped with a smooth endomorphism. Complex best rterm approximations almost always exist in. An ordinary almost complex structure is a choice of a halfdimensional subspace of each fiber of the complexified tangent bundle tm. A vector bundle has the local appearance of a product, but may have nontrivial global topology.
Cr structures on open manifolds rutgers university. Given two complex manifolds m1 and m2 a morphism f. Moreover x is almost homogeneous, with automorphisms induced. In other words, we have a smooth tensor field j of degree 1, 1 such that. Contact manifolds and generalized complex structures. The complex structure of m defines an almost complex structure j. Geometry of twistor spaces claude lebrun simons workshop lecture, 73004 lecture notes by jill mcgowan. If eis a complex vector bundle over mand, is a hermitian. The pair m,j is called an almost complex manifold, and whenever j arises from 1.
We discuss the hodge theory for hermitian bundles over compact almost complex manifolds in details and are able to show theorem 1. Almost complex structure on a real vector space let v be a. Almost complex manifolds almost complex structures. Keywords riemannian manifolds, almost hermitian and almost hyper hermitian structures, tangent bundle 1. A bundle over a manifold is trivial if it is simply the cartesian product of. Let l be a hermitian line bundle endowed with a hermitian connection rl and let e be a hermitian vector bundle over m endowed with an hermitian connection re. I am writing because i am extremely confused with the structure of complex vector bundles. An almost complex structure on f is a complex isomorphism. Strictly speaking, a vector bundle is an entire triple e. Tm is simply what we call an almost complex structure on m. Part of chapter 2, introducing ktheory, then proving bott periodicity in the complex case and adams theorem on the hopf invariant, with its famous applications to division algebras and parallelizability of spheres. More generally, an almost complex structure on a real di. Okumura, on the almost complex structure of tangen bundles of riemannian spaces. It is important to note that, given a topological or smooth vector bundle e, holomorphicity.
Recall that a smooth manifold m is said to be weakly almost complex if its tangent bundle xm is stably isomorphic to the realification of a complex vector bundle over m. Find materials for this course in the pages linked along the left. Almost complex and hermitian vector spaces and vector bundles a let jbe an almost complex structure on a 2ndimensional real vector space v i. In this paper, the standard almost complex structure on the tangent bunle of a riemannian manifold will be generalized. We can find similar situation in mathematical analysis real and complex. Almost hermitian structures on tangent bundles hiroyasu satoh abstract in this article, we consider the almost hermitian structure on tm induced by a pair of a metric and an a.
Some relationships between the geometry of the tangent bundle and the geometry of the riemannian base manifold henry, guillermo and keilhauer, guillermo, tokyo journal of mathematics, 2012. The tangent bundle of an almost complex free loopspace morava, jack, homology, homotopy and applications, 2001. Every positive vector bundle in the sense of grauert is almost positive. When m is regarded as a real manifold, tm is a real vector bundle of rank 2n. Remarks on some almost hermitian structure on the tangent bundle, ii oguro, takashi and takashima, yoshitsugu, nihonkai mathematical journal, 2014.
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